1 Motivation and Background

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1 1 Motivation and Background Figure 1: Periodic orbits of the Reeb vector field on S 3 parametrized by S 2, aka the Hopf fibration (credit: Niles Johnson). My research is in symplectic and contact geometry with a focus on moduli problems in contact topology. I have been working to provide foundations and computations of contact homology invariants. These Floertheoretic invariants are closely related to symplectic homology and have applications to singularity theory, dynamics, string theory, mirror symmetry, knot theory, and symplectic embedding problems. My research combines techniques from differential topology and geometry, partial differential equations, complex geometry, and algebraic topology and geometry. Symplectic and contact structures first arose in the study of classical mechanical systems such as those describing planetary motion and wave propagation; see [Ar78, Ne16]. Solutions of these systems are flow lines of either the Hamiltonian vector field on a symplectic manifold or the Reeb vector field on a contact manifold, see Figure 1. Understanding the evolution and distinguishing transformations of these systems necessitated the development of global invariants of symplectic and contact manifolds which encode structural aspects of the Reeb and/or Hamiltonian flows. These invariants are built out of moduli spaces of pseudoholomorphic curves [Gr85], which are solutions to a nonlinear Cauchy-Riemann equation. They originated from Floer s breakthrough [Fl88] to combine classical variational methods, Gromov s theory of pseudoholomorphic curves, and Witten s interpretation of Morse theory [Wi82]. Formally, a contact structure ξ is a maximally nonintegrable hyperplane distribution. This is in contrast to an integrable hyperplane distribution, whose hyperplanes are given by the tangent spaces of a submanifold; see Figure 2. Any 1-form λ whose kernel defines a contact structure is called a contact form. The Reeb vector field R λ depends on the Figure 2: Integrable vs. contact structures on R 3. choice of contact form λ and is defined by λ(r λ ) = 1, dλ(r λ, ) = 0. A closed Reeb orbit of period T > 0 is defined modulo reparametrization by: γ : R/TZ M, γ(t) = R λ (γ(t)), (1) and is said to be simple, or equivalently embedded, whenever (1) is injective. The linearized Reeb flow for time T defines a symplectic linear map of (ξ, dλ) and if this map does not have 1 as an eigenvalue then γ is said to be nondegenerate. The contact form λ is called nondegenerate if all Reeb orbits are nondegenerate; generic contact forms have this property. 2 Overview of results Constructions of moduli based invariants looked promising with the advent of a comprehensive symplectic field theory, announced in 2000 by Eliashberg, Givental, and Hofer [EGH00]. Symplectic field theory generalizes Floer and Gromov-Witten theories by studying moduli spaces of pseudoholomorphic curves from punctured Riemann surfaces to certain noncompact symplectic manifolds with asymptotics on periodic orbits of the Hamiltonian or Reeb vector field. There is a Fredholm theory describing these moduli spaces as the zero set of an infinite dimensional bundle. To obtain contact and symplectic invariants one must first regularize these moduli spaces so that the (compactified) moduli spaces can be given the structure of a smooth manifold or orbifold, possibly with boundary and corners. When multiply covered curves are present, as in symplectic field theory, there is typically a failure of transversality of the zero set which describes the moduli space these curves. The most 1

2 general approach to regularize these moduli spaces is to abstractly perturb the standard nonlinear Cauchy-Riemann equation. However, the use of abstract perturbations is a lengthy technical endeavor which is often not suitable for applications and in some cases not well understood by the community. My research, in part joint with Hutchings, makes use of more direct methods to extend the transversality theory for the standard pseudoholomorphic curve equation and to isolate precise phenomena which can be accounted for with established geometric and analytic methods. In particular, we have provided foundations for a subset of symplectic field theory known as (cylindrical) contact homology, whose chain complex is generated by certain closed Reeb orbits and whose differential counts pseudoholomorphic cylinders interpolating between these Reeb orbits, by extending the methods of my thesis [Ne13, Ne15]. Our work [HN1, HN2, HN3] gives a rigorous definition of cylindrical contact homology for dynamically convex contact forms in three dimensions (cf. Definition 3.1), invariance in the dynamically convex case, a definition of local contact homology in any dimension, and a substitute for cylindrical contact homology in higher dimensions in the absence of contractible Reeb orbits. These foundational results were not previously available in the literature, cf. [Bo02, Bo09, BO09, BCE07, Us99, MLY04]. Section 2.1 gives an overview of my foundational results with Hutchings. My work on applications of contact invariants and my supervision of associated undergraduate research projects can be found in Section Foundational results Cylindrical contact homology is in principle an invariant of contact manifolds (Y, ξ) that admit a nondegenerate contact form λ without Reeb orbits of certain gradings. The cylindrical contact homology of (Y, ξ) is defined by choosing a nondegenerate contact form λ, taking the homology of a chain complex over Q which is generated by good Reeb orbits and denoted by CC Q (Y, λ, J), and whose differential Q counts J-holomorphic cylinders in R Y for a suitable almost complex structure J. The grading on the complex is given by the Conley-Zehnder index, a winding number associated to the path of symplectic matrices obtained from linearizing the flow along γ, restricted to ξ. The Conley-Zehnder index is denoted by CZ τ (γ) and typically depends on a choice of trivialization τ. Unfortunately, in many cases there is no way to choose J so as to obtain the transversality for holomorphic cylinders needed to define Q, to show that ( Q ) 2 = 0, and to prove that the homology is invariant of the choice of J and λ. In [HN1] we extended the methods from my thesis [Ne13, Ne15] and showed that the cylindrical contact homology differential Q can be defined by directly counting pseudoholomorphic cylinders in dimension 3 for the class of dynamically convex contact forms, cf. Definition 3.1 without any abstract perturbation of the Cauchy-Riemann equation. Our work [HN2, HN3] establishes the invariance of cylindrical contact homology under choices of J and λ. Theorem 2.1. [HN1, HN2, HN3] Let λ be a nondegenerate, dynamically convex contact form on a closed three-manifold Y. Suppose further that: (*) A contractible Reeb orbit γ has CZ(γ) = 3 only if γ is embedded. Then for generic λ-compatible almost complex structures J on R Y, (CC Q (Y, λ, J), Q ), is a well-defined chain complex and its homology is invariant under choices of dynamically convex λ defining the contact structure ξ and generic J, CH Q (Y, ξ) := H (CC Q (Y, λ, J), Q ). Under the assumptions of Theorem 2.1 we also define the following related contact homology theories. Further details are provided in Section 3.1. Theorem 2.2. [HN2, HN3] Under the assumptions of Theorem 2.1, (i) There is a well-defined nonequivariant contact homology defined with coefficients in Z which is a contact invariant and denoted by NCH. 2

3 (ii) There is a well-defined equivariant contact homology defined with coefficients in Z which is a contact invariant and denoted by CH Z. (iii) There is a canonical isomorphism, CH Q CH Z Q. As a result of Theorem 2.2(iii) we call CH Z an integral lift of cylindrical contact homology. Our constructions also produce a definition of local contact homology in any dimension, a theory which has important applications in dynamics [GG09, HM15]. Long term dynamical applications include existence results for homoclinic orbits in Hamiltonian systems [HW90], the study of periodic orbits of iterated disk maps [BrHof12], and the relationship between the topology of a contact 3-manifold and the minimal number of simple Reeb orbits. Our work is connected to several other contact and symplectic invariants. In particular, the nonequivariant and integral theories are expected to be isomorphic 1 to positive symplectic homology and positive S 1 - equivariant symplectic homology, respectively. Symplectic homology is a Floer type invariant of symplectic manifolds with contact type boundary which detects the symplectic structure of the interior and the Reeb dynamics of the boundary [CFHW, Se08, Vi99]. The pos- Figure 3: Two views of the right handed Legendrian trefoil in (R 3, ξ std ), courtesey of Lenny Ng. itive symplectic homology complex is a quotient complex of total complex by the complex generated by critical points in the interior of the symplectic manifold. We also expect that our work should allow for further progress on the classification of Legendrian knots up to Legendrian isotopy in closed dynamically convex contact 3-manifolds. Legendrian knots are smooth knots whose tangent vectors live in the contact structure, see Figure 3 and Legendrian isotopic are those which are isotopic through a family of Legendrian knots. The classification of Legendrian knots is an interesting problem because Legendrian knots are surgery loci for constructions of new contact manifolds and have close ties with smooth knots via geometric and quantum knot invariants. Connections with our work to symplectic homology [BO] and Legendrian contact homology [BEE12, EES07, EENS13] are included in Section 3.2. Additionally, there are the following alternate approaches to defining contact homology invariants. Remark 2.3. (On related approaches) (Alternatives to CH Q ) Bao and Honda [BaHon1] give a definition of a contact invariant akin to cylindrical contact homology for contact manifolds which admit no contractible Reeb orbits in dimension 3, defined with coefficients in Q. It remains to show this definition agrees with the classical one from [EGH00]. Gutt [Gu] has shown that the positive equivariant portion of symplectic homology is also an invariant for some contact manifolds, see also Remark 3.3. (Kuranishi) Pardon s approach [Pa] to defining full contact homology via virtual fundamental cycles yields a definition of cylindrical contact homology in the absence of contractible Reeb orbits. Bao and Honda [BaHon2] give a definition of the full contact homology differential graded algebra for any closed contact manifold in any dimension. These approaches make use of Kuranishi structures to construct contact and symplectic invariants and while they hold more generally, they are also more abstract and hence more difficult to work with in computations and applications; see also [FO99], [FO 3 1]-[FO 3 4], [IP], [Joy], [MW1]-[MW3], [Pa16], [TZ1], [TZ2]. (Polyfolds) Hofer, Wysocki, and Zehnder have developed the abstract analytic framework [HWZI]- [HWZIII], [HWZ-gw], collectively known as polyfolds, to systematically resolve issues of regularizing moduli spaces. Contact homology awaits foundations via polyfolds and the use of 1 See Section 3.2 for a more precise explanation. 3

4 abstract perturbations can make computations difficult. Hofer and I plan to investigate the connections between the polyfold approach and the one taken by Hutchings and myself. 2.2 Computational methods and undergraduate research supervision I am currently applying the computational methods I developed in [Ne13, Ne15, Ne] to prove a Floer theoretic interpretation of the McKay correspondence [IM96] of the links of the simple singularities. The simple singularities can be characterized as the absolutely isolated double point quotient singularity of C 2 /Γ, where Γ is a finite subgroup of SL(2; C). The link of these singularities is (contact) diffeomorphic to S 3 /Γ. Theorem in Progress 2.4. The cylindrical contact homology of the link of a simple singularity is a free Q[u] module of rank equal to the number of conjugacy classes of the respective finite subgroup Γ of SL(2; C). My method of proof demonstrates the cohomological McKay correspondence via the Reeb dynamics of these links. This is related to work in progress by McLean and Ritter, who are formulating a McKay correspondence for links of crepant singularities in terms of symplectic homology. Further details on these computational methods as well as their expected extension to more general Seifert fiber spaces and connections with Chen-Ruan cohomology and string topology may be found in Section 4. Many aspects of computing Floer theoretic invariants can be made appropriate research projects for graduate students and talented undergraduates. I supervised an REU project over two summers resulting in the verification of the case of the A n singularity via direct computations [AHNS]. Our approach involved the construction of an explicit contact form on L(n + 1, n) which admitted exactly two simple Reeb orbits and computed the associated Conley-Zehnder indices of these orbits and their iterates. In recent work with Christianson [CN], a former REU mentee, we found new obstructions to symplectic embeddings of the four-dimensional polydisk, P(a, 1) = {(z 1, z 2 ) C 2 π z 1 2 a, π z 2 2 1}, into the ball, B(c) = { (z 1, z 2 ) C 2 π z π z 2 2 c }, extending work done by Hind-Lisi [HL15] and Hutchings [Hu16]. We made use of Hutchings refinement of embedded contact homology from [Hu16], which established a necessary condition for one convex toric domain, such as an ellipsoid or polydisk, to symplectically embed into another. Theorem 2.5 ([CN]). Let 2 a < = If the four dimensional polydisk P(a, 1) symplectically embeds into the four dimensional ball B(c) = E(c, c) then c 2 + a 2. Schlenk s folding construction permits us to conclude our bound on c is optimal, [Sc15, Prop ]. Additionally, we proved that if certain symplectic embeddings of four dimensional convex toric domains exist then a modified version of the combinatorial criterion from [Hu16] must hold, thereby reducing the computational complexity of the original criterion from O(2 n ) to O(n 2 ). 3 Technical aspects of contact homology and related invariants This section provides further details on Theorems 2.1 and 2.2, in regards to the aforementioned class of dynamically convex class of contact forms. Definition 3.1. (cf. [HWZ99]) Let λ be a nondegenerate contact form on a closed three-manifold Y. We say that λ is dynamically convex if either: λ has no contractible Reeb orbits, or 4

5 c 1 (ξ) π2 (Y) = 0, and every contractible Reeb orbit γ has CZ(γ) 3. Generic convex, compact, star-shaped hypersurfaces Y in R 4 admit a dynamically convex contact form via the restriction of λ = k=1 (x kdy k y k dx k ) to Y. Theorem 2.1 required the following additional assumption, which we briefly explain. Remark 3.2 (On the hypotheses of Theorem 2.1 and related future work). (a) The hypothesis (*) automatically holds when π 1 (Y) contains no torsion: (*) A contractible Reeb orbit γ has CZ(γ) = 3 only if γ is embedded, (b) In general, the hypothesis (*) can be removed from Theorem 2.1 assuming a certain technical conjecture on the asymptotics of holomorphic curves. (c) We expect that the hypothesis of dynamical convexity can be further weakened. 3.1 Discussion of Theorem 2.2 Our work [Ne13, Ne15, HN1] demonstrates that generic S 1 -independent almost complex structures yield sufficient transversality to ensure that Q is well-defined and that ( Q) 2 = 0 for large classes of contact manifolds in dimension 3. However, S 1 -independent almost complex structures do not give sufficient transversality to count index zero cylinders in cobordisms, which is necessary to define the chain maps. This is because the covers of index zero cylinders can live in a moduli space of negative virtual dimension. In order to prove topological invariance of CH Q (Y, λ, J) in [HN2, HN3] we make use of S 1 -dependent almost complex structures, similar to the set up in [BO]. Breaking the S 1 symmetry invalidates certain properties needed to prove ( Q) 2 = 0 and the chain map and chain homotopy equations. As a result, the use of S 1 -dependent almost complex structures leads to a Morse-Bott version of the chain complex. The homology of this chain complex is not the desired cylindrical contact homology but rather a nonequivariant version of it. This nonequivariant version is denoted by NCH and the content of Theorem 2.2(i). Moreover, the presence of contractible Reeb orbits necessitates the use of obstruction bundle gluing [HT07, HT09], producing a correction term in the expression of the nonequivariant differential [HN3]. The desired cylindrical contact homology CH Q can be regarded as an S 1 - equivariant version of nonequivariant contact homology, and recovering this [HN2, HN3] requires additional family Floer theoretic constructions in the spirit of [BO, Hu08, SeSm10]. An outcome of these constructions is our so called integral lift of contact homology, denoted by CH Z, and the content of Theorem 2.2(ii)-(iii). We have examples which show that this integral lift of contact homology also contains interesting torsion information pertaining to the qualitative behavior of the Reeb dynamics. The entirety of Theorem 2.2 allows us to conclude that CH Q is a contact invariant. 3.2 Connections with symplectic homology and Legendrian contact homology The preprint of Bourgeois and Oancea [BO] suggests that an isomorphism between the positive part of S 1 -equivariant symplectic homology and linearized contact homology should exist in general. Their proof requires the assumption that an almost complex structure J can be chosen so that all relevant moduli spaces of J-holomorphic curves are cut out transversely. 2 Our work indicates that the construction of a geometric isomorphism must involve an obstruction bundle contribution term in the presence of contractible orbits. We plan to pursue this avenue of study in the coming year, and expect an affirmative answer to the following conjecture. 2 This assumption is only true for contact manifolds arising as unit cotangent bundles DT L with dim L 5 or those Riemannian manifolds L which admit no contractible closed geodesics. 5

6 Conjecture 1. Under the assumptions of Theorem 2.1 there exist natural isomorphisms, NCH (Y, ξ) SH + (R Y, ξ; Z); CH Z (Y, ξ) SH +,S1 (R Y, ξ; Z). (2) Remark 3.3. More precisely, NCH (Y, ξ) should be isomorphic to the positive symplectic homology SH + of a filling of Y. In fact, for a contact manifold with no contractible Reeb orbits, or more generally for a contact manifold of dimension 2n 1 in which every contractible Reeb orbit satisfies CZ(γ) > 4 n, one can define positive symplectic homology directly in terms of the symplectization, without reference to a filling [BO, Section 4.1.2(2)]; this is satisfied by the dynamically convex assumption in dimension 3. This positive symplectic homology (and its equivariant version) is shown to be an invariant of the closed contact manifold in [Gu, Theorems 1.2, 1.3]. Additionally we expect that our work can be incorporated into the framework of Legendrian contact homology [BEE12, CELN, EES07, EENS13]. This relative version of contact homology is currently defined for Legendrians in 1-jet spaces [EENS13] and for Legendrian links in # k (S 1 S 2 ) [EN15]. We anticipate that our results will permit this theory to be geometrically defined for Legendrian knots in closed dynamically convex contact 3-manifolds and that the proof of the following conjecture is within reach of our methods. Conjecture 2. The contact homology of Legendrian knots of dynamically convex contact 3-manifolds is welldefined: The stable tame isomorphism class of the DGA associated to a Legendrian knot K is independent of the choice of compatible almost complex structure J and is invariant under Legendrian isotopies of K. There is also the following hope that one can relate the Legendrian DGA to other combinatorial knot invariants as in [ENS, EENS13]. Such a knot invariant in combination with a combinatorial expression would lead to progress on the following question. Question 3. Do there exist infinite families of Legendrian knots in dynamically convex contact 3-manifolds, which have the same classical invariants but are pairwise non Legendrian isotopic? 4 Futher details on McKay correspondence and computations There is a canonical contact form α 0 associated to prequantization spaces [BW58] and certain S 1 -bundles over symplectic orbifolds Y; this form is is degenerate, meaning that the closed Reeb orbits are non-isolated. We can directly perturb the critical manifolds realized as Reeb orbits with a Morse-Smale function H on the base B that is invariant under the S 1 - action of the bundle S 1 Y h B. The perturbation, yields the Reeb dynamics, α ɛ = (1 + ɛh H)α 0, R ɛ = (1 + ɛh H) 1 R + ɛ(1 + ɛh H) 2 X H, where X H is a Hamiltonian vector field on S 2 and X H its horizontal lift. The only fibers that persist as Reeb orbits of R ɛ are those over the Figure 4: H for H = z with a fiber over S 2, respectively S 2 /Z 3. critical points of H. Additional Reeb orbits of R ɛ must come from horizontal lifts of closed orbits of X H. Since ɛh and ɛdh are small, these Reeb orbits all have action (e.g. length) greater than 1/ɛ. Thus the only closed Reeb orbits left intact by this perturbation up must occur as a multiple cover of a simple Reeb orbit lying over a critical point of H. There is a proportionality between the action of these persisting orbits and their length. This filtration on both the action and index leads to a 6

7 formal version of filtered contact homology, allowing one to recover cylindrical contact homology in a well-defined way [Ne13, Ne15]. In particular, the only orbits which generate the filtered chain complex must project to critical points on the base. The height function on S 2 yields the desired computation of CH Q (S 3, ξ std ), see Figure 4. The simple Reeb orbit projecting to the north pole has index 2 and the one projecting to the south pole has index 0. The index increases by 4 after each iteration of the Reeb orbit. 3. We thus recover the well known computation of cylindrical contact homology for the sphere (S 3, ξ std ). Example 4.1 ([Ne13]). The cylindrical contact homology for the sphere (S 3, ξ std ) is given by { CH Q (S 3 Q 2, even;, ξ std ) = 0 else. If we consider the lens space (L(n + 1, n), ξ std ) with its contact structure induced by the one on S 3, then the free homotopy classes of the orbits of R ɛ over critical points of H are in 1-1 correspondence with the conjugacy classes of L(n + 1, n). While (*) from Theorem 2.1 is not satisfied, the methods of [Ne15] still allow us to conclude that we get a well-defined chain complex which is invariant under the choice of contact form and almost complex structure. Theorem 4.2 ([Ne]). The cylindrical contact homology for the sphere (L(n + 1, n), ξ std ) is given by Q n+1 2, even; CH Q ((L(n + 1, n), ξ std ) = Q n = 0; 0 else. This computation agrees with what is obtained in [AHNS] for the link of the A n singularity, which is to be expected given that these two manifolds are contactomorphic. The differential Q should behave as follows for perturbations of α 0 by H different from the height function. Theorem in Progress 4.3. The pseudoholomorphic cylinders counted by the cylindrical contact homology differential lie over Morse-Smale trajectories of H in the base. The simple singularities appear as the origin of the variety C 2 /Γ, where Γ SU 2 (C) is a finite subgroup. This perturbation lends itself to a means of computing the cylindrical contact homology of the links of the simple singularities [Ne], viewed as the quotient spaces (S 3 /Γ, ξ can ). In this set up, we must pick H so that X H is Figure 5: H and X H = J 0 H invariant under the image of Γ in SO(3). In the case of the E 6 singularity Γ = T is the binary tetrahedral group and taking H = xyz yields the desired T- invariant vector field X H on S 3, as in Figure 5. After quotienting out the entire bundle by the action of Γ, the free homotopy classes of the orbits of R ɛ over critical points of H are in 1-1 correspondence with the conjugacy classes of Γ. As a result, the rank of CH Q is governed by the number of these conjugacy classes, producing the McKay correspondence theorem mentioned in Section 2.2. Moreover, this perturbed Reeb flow realizes the presentation of the spherical manifolds S 3 /Γ as Seifert fiber spaces. We obtain S 1 -bundles over the 2-sphere with the expected number of exceptional fibers in 1-1 correspondence with the generators of the chain complex. These methods should extend to other Seifert spaces, resulting in a connection with Chen-Ruan cohomology [CR04], and suggesting a generalization of Ritter s recent work [Ri14] on Floer theory for negative line bundles via Gromov-Witten theory of orbifolds. 3 A computation for the grading yields γ k p = CZ(γ k p) 1 = 4k 2 + index p (H). Here p Crit(H), and γ k p is the k-fold iterate of a simple Reeb orbit γ over p. 7

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